He continued with an example:
‘Suppose a friend states that he has mislaid a key somewhere in his house and asks you to help him find it. If you believe his memory to be faultless and you have absolute trust in his integrity, what does it mean?’
‘It means that he has indeed mislaid the key somewhere in his house.’
‘And if he further ascertains that no one else entered the house since?’
‘Then we can assume that it was not taken out of the house.’
‘Ergo?’
‘Ergo, the key is still there, and if we search long enough — the house being finite — sooner or later we will find it.’
My uncle applauded. ‘Excellent! It is precisely this certainty that fuelled my optimism anew. After I had recovered from my first disappointment I got up one fine morning and said to myself: “What the hell — that proof is still out there, somewhere!”’
‘And so?’
‘And so, my boy, since the proof existed, one had but to find it!’
I wasn’t following his reasoning.
‘I don’t see how this provided comfort, Uncle Pet-ros: the fact that proof existed didn’t in any way imply that you would be the one to discover it!’
He glared at me for not immediately seeing the obvious. ‘Was there anyone in the whole wide world better equipped to do so than I, Petros Papachristos?’
The question was obviously rhetorical and so I didn’t bother to answer it. But I was puzzled: the Petros Papachristos he was referring to was a different man from the self-effacing, withdrawn senior citizen I’d known since childhood.
Of course, it had taken him some time to recover from reading Hardy’s letter and its disheartening news. Yet recover he eventually did. He pulled himself together and, his deposits of hope refilled through the belief in ‘the existence of the proof somewhere out there’, he resumed his quest, a slightly changed man. His misadventure, by exposing an element of vanity in his manic search, had created in him an inner core of peace, a sense of life continuing irrespective of Goldbach’s Conjecture. His working schedule now became slightly more relaxed, his mind also aided by interludes of chess, more tranquil despite the constant effort.
In addition, the switch to the algebraic method, already decided in Innsbruck, made him feel once again the excitement of a fresh start, the exhilaration of entering virgin territory.
For a hundred years, from Riemann’s paper in the mid-nineteenth century, the dominant trend in Number Theory had been analytic. By now resorting to the ancient, elementary approach, my uncle was in the vanguard of an important regression, if I may be allowed the oxymoron. The historians of mathematics will do well to remember him for this, if for no other part of his work.
It must be stressed here that, in the context of Number Theory, the word ‘elementary’ can on no account be considered synonymous with ‘simple’ and even less so with ‘easy’. Its techniques are those of Diophantus’, Euclid’s, Fermat’s, Gauss’s and Euler’s great results and are elementary only in the sense of deriving from the elements of mathematics, the basic arithmetical operations and the methods of classical algebra on the real numbers. Despite the effectiveness of the analytic techniques, the elementary method stays closer to the fundamental properties of the integers and the results arrived at with it are, in an intuitive way clear to the mathematician, more profound.
Gossip had by now seeped out from Cambridge, that Petros Papachristos of Munich University had had a bit of bad luck, deferring publication of very important work. Fellow number theorists began to seek his opinions. He was invited to their meetings, which from that point on he would invariably attend, enlivening his monotonous lifestyle with occasional travel. The news had also leaked out (thanks here to the Director of the School of Mathematics) that he was working on the notoriously difficult Conjecture of Goldbach, and that made his colleagues look on him with a mixture of awe and sympathy.
At an international meeting, about a year after his return to Munich, he ran across Littlewood. ‘How’s the work going on Goldbach, old chap?’ he asked Petros.
‘Always at it.’
‘Is it true what I hear, that you’re using algebraic methods?’
‘It’s true.’
Littlewood expressed his doubts and Petros surprised himself by talking freely about the content of his research. ‘After all, Littlewood,’ he concluded, ‘I know the problem better than anyone else. My intuition tells me the truth expressed by the Conjecture is so fundamental that only an elementary approach can reveal it.’
Littlewood shrugged. ‘I respect your intuition, Papachristos; it’s just that you are totally isolated. Without a constant exchange of ideas, you may find yourself grappling with phantoms before you know it.’
‘So what do you recommend,’ Petros joked, ‘issuing weekly reports of the progress of my research?’
‘Listen,’ said Littlewood seriously, ‘you should find a few people whose judgement and integrity you trust. Start sharing; exchange, old chap!’
The more he thought about this suggestion, the more it made sense. Much to his surprise he realized that, far from frightening him, the prospect of discussing the progress of his work now filled him with pleasurable anticipation. Of course his audience would have to be small, very small indeed. If it was to consist of people ‘whose judgement and integrity he trusted’, that would of necessity mean an audience of no more than two: Hardy and Littlewood.
He started anew the correspondence with them that he’d interrupted a couple of years after he left Cambridge. Without stating it in so many words, he dropped hints about his intention to bring about a meeting during which he would present his work. Around Christmas of 1931, he received an official invitation to spend the next year at Trinity College. He knew that since, for all practical purposes, he had been absent from the mathematical world for a long long time, Hardy must have used all his influence to secure the offer. Gratitude, combined with the exciting prospect of a creative exchange with the two great number theorists, made him immediately accept.
Petros described his first few months in England, in the academic year 1932-33, as probably the happiest of his life. Memories of his first stay there, fifteen years earlier, infused his days at Cambridge with the enthusiasm of early youth, as yet untainted by the possibility of failure.
Soon after he arrived, he presented to Hardy and Littlewood the outline of his work to date with the algebraic method, and this gave him the first taste, after more than a decade, of the joy of peer recognition. It took him several mornings, standing at the blackboard in Hardy’s office, to trace his progress in the three years since his volte-face from the analytic techniques. His two renowned colleagues, who were at first extremely sceptical, now began to see some advantages to his approach, Littlewood more so than Hardy.
‘You must realize,’ the latter told him, ‘that you’re running a huge risk. If you don’t manage to ride this approach to the end, you’ll be left with precious little to show for it. Intermediate divisibility results, although quite charming, are not of much interest any more. Unless you can convince people that they can be useful in proving important theorems, like the Conjecture, they are not of themselves worth much.’
Petros was, as always, well aware of the risks he was taking.
‘Still, something tells me you may well be on a good course,’ Littlewood encouraged him.
‘Yes,’ grumbled Hardy, ‘but please do hurry up, Papachristos, before your mind begins to rot, the way mine’s doing. Remember, at your age Ramanujan was already five years dead!’
This first presentation had taken place early in the Michaelmas term, yellow leaves falling outside the Gothic windows. During the winter months that followed, my uncle’s work advanced more than it ever had. It was at this time that he also started using the method he called ‘geometric’.
He began by representing all composite (i.e. non-prime) numbers by placing dots in a parallelogram, with the lowest prime divisor as width
and the quotient of the number by it as height. For example, 15 is represented by 3 × 5 rows, 25 by 5 × 5, 35 by 5 × 7 rows:
By this method, all even numbers are represented as double columns, as 2 × 2, 2 × 3, 2 × 4, 2 × 5, etc.
The primes, on the contrary, since they have no integer divisors, are represented as single rows, for example 5, 7, 11:
Petros extended the insights from this elementary geometric analogy to arrive at number-theoretical conclusions.
After Christmas, he presented his first results. Since, however, instead of using pen and paper, he laid out his patterns on the floor of Hardy’s study using beans, his new approach earned from Littlewood a teasing accolade. Although the younger man conceded that he found ‘the famous Papachristos bean method’ conceivably of some usefulness, Hardy was by now outright annoyed.
‘Beans indeed!’ he said. ‘There is a world of difference between elementary and infantile … Don’t you forget it, Papachristos, this blasted Conjecture is difficult — if it weren’t, Goldbach would have proved it himself!’
Petros, however, had faith in his intuition and attributed Hardy’s reaction to the ‘intellectual constipation brought about by age’ (his words).
‘The great truths in life are simple,’ he told Little-wood later, when the two of them were having tea in his rooms. Littlewood countered him, mentioning the extremely complex proof of the Prime Number Theorem by Hadamard and de la Vallée-Poussin.
Then he made a proposal: ‘What would you say to doing some real mathematics, old chap? I’ve been working for some time now on Hilbert’s Tenth Problem, the solvability of Diophantine equations. I have this idea that I want to test, but I’m afraid I need help with the algebra. Do you think you could lend me a hand?’
Littlewood would have to seek his algebraic help elsewhere, however. Although his colleague’s confidence in him was a boost to Petros’ pride, he flatly declined. He was too exclusively involved with the Conjecture, he said, too deeply engrossed in it, to be able fruitfully to concern himself with anything else.
His faith, backed by a stubborn intuition, in the ‘infantile’ (according to Hardy) geometric approach, was such that for the first time since he began work on the Conjecture, Petros now often had the feeling that he was almost a hair’s breadth away from the proof. There were actually even a few exhilarating minutes, late on a sunny January afternoon, when he had the short-lived illusion that he had succeeded — but, alas, a more sober examination located a small, but crucial mistake.
(I have to confess it, dear reader: at this point in my uncle’s narrative I felt despite myself a quiver of vengeful joy. I remembered that summer in Pylos, a few years back, when I too had thought for a while I’d discovered the proof of Goldbach’s Conjecture — although I did not then know it by name.)
His great optimism notwithstanding, Petros’ occasional bouts of self-doubt, sometimes verging on despair (especially after Hardy’s put-down of the geometric method), now became stronger than ever. Still, they could not curb his spirit. He fought them away by branding them the inevitable anguish preceding a great triumph, the onset of the labour pains leading into the delivery of the majestic discovery. After all, the night is darkest before dawn. He was, Petros felt certain, all but ready to run the final dash. One last concentrated burst of effort was all that was needed to award him the last brilliant insight.
Then, there would come the glorious finish …
The heralding of Petros Papachristos’ surrender, the termination of his efforts to prove Goldbach’s Conjecture, came in a dream he had in Cambridge, sometime after Christmas — a portent whose full significance he did not at first fathom.
Like many mathematicians working for long periods with basic arithmetical problems, Petros had acquired the quality that has been called ‘friendship with the integers’, an extended knowledge of the idiosyncrasy, quirks and peculiarities of thousands of specific whole numbers. A few examples: a ‘friend of the integers’ will immediately recognize 199 or 457 or 1009 as primes. 220 he will automatically associate with 284 since they are linked by an unusual relationship (the sum of the integer divisors of each one is equal to the other). 256 he reads naturally as 2 to the eighth power, which he well knows to be followed by a number with great historical interest, since 257 can be expressed as and a famous hypothesis held that all numbers of the form were prime.*
The first man my uncle had met who had this quality (and to the utmost degree) was Srinivasa Ramanujan. Petros had seen it demonstrated on many opportunities, and to me he recounted this anecdote:†
One day in 1918, he and Hardy were visiting him in the sanatorium where he lay ill. To break the ice, Hardy mentioned that the taxi that had brought them had had the registration number 1729, which he personally found ‘rather boring’. But Ramanujan, after pondering this for only a moment, disagreed vehemently: ‘No, no, Hardy! It’s a particularly interesting number — in fact, it’s the smallest integer that can be expressed as the sum of two cubes in two different ways!’*
During the years that Petros worked on the Conjecture with the elementary approach, his own friendship with the integers developed to an extraordinary degree. Numbers ceased after a while being inanimate entities; they became to him almost alive, each with a distinct personality. In fact, together with the certainty that the solution existed somewhere out there, it added to his resolve to persevere during the most difficult of times: working with the integers, he felt, to quote him directly, ‘constantly among friends’.
This familiarity caused an influx of specific numbers into his dreams. Out of the nameless, nondescript mass of integers that up until then crowded their nightly dramas, individual actors now began to emerge, even occasional protagonists. 65, for example, appeared for some reason as a City gentleman, with bowler hat and rolled umbrella, in constant companionship with one of his prime divisors, 13, a goblinlike creature, supple and lightning-quick. 333 was a fat slob, stealing bites of food from the mouths of its siblings 222 and in, and 8191, a number known as a ‘Mersenne Prime’, invariably wore the attire of a French gamin, complete down to the Gauloise cigarette hanging from his lips.
Some of the visions were amusing and pleasant, others indifferent, still others repetitious and annoying. There was one category of arithmetical dream, however, which could only be called nightmarish, if not for horror or agony then for its profound, bottomless sadness. Particular even numbers would appear, personified as pairs of identical twins. (Remember that an even number is always of the form 2k, the sum of two equal integers). The twins would gaze on him fixedly, immobile and expressionless. But there was great, if mute, anguish in their eyes, the anguish of desperation. If they could have spoken, their words would have been: ‘Come! Please. Hurry! Set us free!’
It was a variation on these sad apparitions that came to wake him one night late in January 1933. This was the dream that he termed in retrospect ‘the herald of defeat’.
He dreamed of 2100 (2 to the hundredth power, an enormous number) personified as two identical, freckled, beautiful dark-eyed girls, looking straight into his eyes. But now there wasn’t just sadness in their look, as there had been in his previous visions of the Evens; there was anger, hatred even. After gazing at him for a long, long while (this in itself was sufficient cause to brand the dream a nightmare) one of the twins suddenly shook her head from side to side with jerky, abrupt movements. Then her mouth was contorted into a cruel smile, the cruelty being that of a rejected lover.
‘You’ll never get us,’ she hissed.
At this, Petros, drenched in sweat, jumped up from his bed. The words that 299 (that’s one half of 2100) had spoken meant only one thing: He was not fated to prove the Conjecture. Of course, he was not a superstitious old woman who would give undue credence to omens. Yet the profound exhaustion of many fruitless years had now begun to take its toll. His nerves were not as strong as they used to be and the dream upset him inordinately.
Unable to go back to sleep, he went o
ut to walk in the dark, foggy streets, to try to shake off its dreary feeling. As he walked in the first light among the ancient stone buildings, he suddenly heard fast footsteps approaching behind him, and for a moment he was seized by panic and turned sharply round. A young man in athletic gear materialized out of the mist, running energetically, uttered a greeting and disappeared once again, his rhythmic breathing trailing off into complete silence.
Still upset by the nightmare, Petros wasn’t sure whether this image had been real, or an overflow of his dream world. When, however, a few months later the very same young man came to his rooms at Trinity on a fateful mission, he instantly recognized him as the early-morning runner. After he was gone, he realized with hindsight that their first, dawn meeting had cryptically signalled the dark forewarning, coming as it did after the vision of 2100, with its message of defeat.
The fatal meeting took place a few months after the first, early-morning encounter. In his diary Petros marks the exact date with a laconic comment — the first and last use of Christian reference I discovered in his diaries: ‘17 March 1933. Kurt Gödel’s Theorem. May Mary, Mother of God, have mercy on me!’
It was late afternoon and he had been in his rooms all day, sitting forward in his armchair studying parallelograms of beans laid out on the floor before him, lost in thought, when there was a knock on the door.
‘Professor Papachristos?’
A blond head appeared. Petros had a powerful visual memory and immediately recognized the young runner, who was full of excuses for disturbing him. ‘Please forgive my barging in on you like this, Professor,’ he said, ‘but I am desperate for your help.’
Petros was quite surprised — he’d thought his presence at Cambridge had gone completely unremarked. He wasn’t famous, he wasn’t even well known and, except at his almost nightly appearances at the university chess club, he hadn’t exchanged two words with anyone other than Hardy and Littlewood during his stay
Uncle Petros and Goldbach's Conjecture Page 9